Sparse operator compression of higher-order elliptic operators with rough coefficients

We introduce the sparse operator compression to compress a self-adjoint higher-order elliptic operator with rough coefficients and various boundary conditions. The operator compression is achieved by using localized basis functions, which are energy-minimizing functions on local patches. On a regular mesh with mesh size $h$, the localized basis functions have supports of diameter $O(h\log(1/h))$ and give optimal compression rate of the solution operator. We show that by using localized basis functions with supports of diameter $O(h\log(1/h))$, our method achieves the optimal compression rate of the solution operator. From the perspective of the generalized finite element method to solve elliptic equations, the localized basis functions have the optimal convergence rate $O(h^k)$ for a $(2k)$th-order elliptic problem in the energy norm. From the perspective of the sparse PCA, our results show that a large set of Mat\'{e}rn covariance functions can be approximated by a rank-$n$ operator with a localized basis and with the optimal accuracy

A sparse decomposition of low rank symmetric positive semi-definite matrices

Suppose that $A \in \mathbb{R}^{N \times N}$ is symmetric positive semidefinite with rank $K \le N$. Our goal is to decompose $A$ into $K$ rank-one matrices $\sum_{k=1}^K g_k g_k^T$ where the modes $\{g_{k}\}_{k=1}^K$ are required to be as sparse as possible. In contrast to eigen decomposition, these sparse modes are not required to be orthogonal. Such a problem arises in random field parametrization where $A$ is the covariance function and is intractable to solve in general. In this paper, we partition the indices from 1 to $N$ into several patches and propose to quantify the sparseness of a vector by the number of patches on which it is nonzero, which is called patch-wise sparseness. Our aim is to find the decomposition which minimizes the total patch-wise sparseness of the decomposed modes. We propose a domain-decomposition type method, called intrinsic sparse mode decomposition (ISMD), which follows the "local-modes-construction + patching-up" procedure. The key step in the ISMD is to construct local pieces of the intrinsic sparse modes by a joint diagonalization problem. Thereafter a pivoted Cholesky decomposition is utilized to glue these local pieces together. Optimal sparse decomposition, consistency with different domain decomposition and robustness to small perturbation are proved under the so called regular-sparse assumption (see Definition 1.2). We provide simulation results to show the efficiency and robustness of the ISMD. We also compare the ISMD to other existing methods, e.g., eigen decomposition, pivoted Cholesky decomposition and convex relaxation of sparse principal component analysis [25] and [40]

AttnGAN: Fine-Grained Text to Image Generation with Attentional Generative Adversarial Networks

In this paper, we propose an Attentional Generative Adversarial Network (AttnGAN) that allows attention-driven, multi-stage refinement for fine-grained text-to-image generation. With a novel attentional generative network, the AttnGAN can synthesize fine-grained details at different subregions of the image by paying attentions to the relevant words in the natural language description. In addition, a deep attentional multimodal similarity model is proposed to compute a fine-grained image-text matching loss for training the generator. The proposed AttnGAN significantly outperforms the previous state of the art, boosting the best reported inception score by 14.14% on the CUB dataset and 170.25% on the more challenging COCO dataset. A detailed analysis is also performed by visualizing the attention layers of the AttnGAN. It for the first time shows that the layered attentional GAN is able to automatically select the condition at the word level for generating different parts of the image

Compressing Positive Semidefinite Operators with Sparse/Localized Bases

Given a positive semidefinite (PSD) operator, such as a PSD matrix, an elliptic operator with rough coefficients, a covariance operator of a random field, or the Hamiltonian of a quantum system, we would like to find its best finite rank approximation with a given rank. One way to achieve this objective is to project the operator to its eigenspace that corresponds to the smallest or largest eigenvalues, depending on the setting. The eigenfunctions are typically global, i.e. nonzero almost everywhere, but our interest is to find the sparsest or most localized bases for these subspaces. The sparse/localized basis functions lead to better physical interpretation and preserve some sparsity structure in the original operator. Moreover, sparse/localized basis functions also enable us to develop more efficient numerical algorithms to solve these problems. In this thesis, we present two methods for this purpose, namely the sparse operator compression (Sparse OC) and the intrinsic sparse mode decomposition (ISMD). The Sparse OC is a general strategy to construct finite rank approximations to PSD operators, for which the range space of the finite rank approximation is spanned by a set of sparse/localized basis functions. The basis functions are energy minimizing functions on local patches. When applied to approximate the solution operator of elliptic operators with rough coefficients and various homogeneous boundary conditions, the Sparse OC achieves the optimal convergence rate with nearly optimally localized basis functions. Our localized basis functions can be used as multiscale basis functions to solve elliptic equations with multiscale coefficients and provide the optimal convergence rate O(hk) for 2k'th order elliptic problems in the energy norm. From the perspective of operator compression, these localized basis functions provide an efficient and optimal way to approximate the principal eigen-space of the elliptic operators. From the perspective of the Sparse PCA, we can approximate a large set of covariance functions by a rank-n operator with a localized basis and with the optimal accuracy. While the Sparse OC works well on the solution operator of elliptic operators, we also propose the ISMD that works well on low rank or nearly low rank PSD operators. Given a rank-n PSD operator, say a N-by-N PSD matrix A (n ≤ N), the ISMD decomposes it into n rank-one matrices Σni=1gigTi where the mode {gi}ni=1 are required to be as sparse as possible. Under the regular-sparse assumption (see Definition 1.3.2), we have proved that the ISMD gives the optimal patchwise sparse decomposition, and is stable to small perturbations in the matrix to be decomposed. We provide several applications in both the physical and data sciences to demonstrate the effectiveness of the proposed strategies.</p